Two-Dimensional Coordinate Geometry (Undergraduate Foundation)

This course provides the complete foundation in Two-Dimensional Coordinate Geometry, establishing the link between algebraic equations and geometric shapes. We systematically cover the algebraic definitions and analysis of straight lines and all four conic sections - the circle, ellipse, parabola, and hyperbola. The curriculum concludes by applying differential calculus to derive tangents and normals to these curves. Command of this geometry is essential for technical modelling in all physical sciences. These equations are used to define component paths in mechanical design and model trajectories in physics. For example, ellipses model orbits, and parabolas model projectile motion and reflector shapes. This course provides the indispensable toolkit for translating visual problems into computable algebraic forms. Upon completion, you will master calculating gradients and distances for straight lines and formulating their equations in various forms. You will derive and manipulate the algebraic equations for all conic sections. Furthermore, you will apply differentiation to find the equations of tangents and normals to any curve at a specified point. This course is mandatory for first-year undergraduate students in Engineering, Physics, and Applied Mathematics. It requires prior knowledge of basic algebra and differential calculus. This programme provides the necessary rigorous foundation in 2D spatial analysis, which is a critical prerequisite for advanced subjects like mechanics and multi-variable calculus.

Enrolment valid for 12 months

This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.

[NUC Core] MTH 103: Elementary Mathematics III - Vectors, Geometry and Dynamics

This comprehensive learning track guides you through the complete world of vector analysis. We begin with the fundamentals of vector algebra and its application to foundational geometry. You will then master scalar, vector, and triple products before using them to construct the vector equations of lines, planes, and conics. The journey culminates in advanced topics, including vector calculus, its applications in classical mechanics, and an introduction to differential geometry. Vectors are the essential language used to describe our physical world, making their mastery non-negotiable for any serious student of science or engineering. This track is designed to build your intuition for spatial reasoning and equip you with a powerful problem-solving toolkit. You will see direct applications in mechanics, analyzing forces and motion; in geometry, calculating angles and distances; and in calculus, modeling dynamic change over time. While this track is tailored to the first-year university curriculum for MTH 104 at Obafemi Awolowo University, Ile-Ife, Nigeria, it is an invaluable resource for a wide range of learners. It is ideal for any undergraduate student in mathematics, physics, engineering, or computer science seeking a comprehensive understanding of vector analysis. Furthermore, it serves as an excellent and thorough refresher for professionals who wish to solidify their foundational knowledge of this critical subject.

Course Chapters

1. Introduction

This foundational chapter provides a concise course outline and establishes the essential framework for all subsequent modules. We review the Cartesian coordinate system, which is the algebraic basis for translating two-dimensional geometry into manipulable equations. This framework is mandatory for successful completion of the course. By the end of this chapter, you will master: identifying the axes and origin in the Cartesian plane; defining points using ordered pairs; and understanding the core necessity of the coordinate system for analytic geometry.

Chapter lessons

1-1. Welcome

10:41

Welcome to the course and outline of course.

1-2. Cartesian coordinates

This lesson reviews the Cartesian coordinate system in the two-dimensional plane. We define the fundamental concepts of axes, origin, and ordered pairs. This is the necessary framework for translating geometric shapes into algebraic equations.

2. Straight Lines

This chapter establishes the algebraic definitions for straight lines in the 2D plane. We cover all standard forms of the line equation. Mastering the line is the essential first step for all subsequent analysis in coordinate geometry, including intersections and conic sections. By the end of this chapter, you will master: calculating gradients and distances; formulating line equations (point-slope, slope-intercept, general); analysing conditions for parallel and perpendicular lines; and finding the intersection point of two lines.

Chapter lessons

2-1. Gradient or slope

This lesson defines the gradient as the measure of a line's steepness and direction in the Cartesian plane. We establish the formula for calculating it using any two points. Mastery of the gradient is essential for defining all subsequent line properties.

2-2. Equation of a line

This lesson derives the three primary algebraic forms for a straight line. Mastering these equations is the fundamental tool for all subsequent problem-solving in the plane.

3. Circles

This chapter introduces the circle as the simplest conic section, establishing its geometric and algebraic definitions. We derive and master the standard and general forms of the equation in the Cartesian plane. Understanding the circle's equation is the foundational step for analysing the more complex conic curves later in the course. By the end of this chapter, you will master: defining the circle using its locus property; deriving the standard and general forms of the equation; identifying the centre and radius from any given equation; and applying these equations to simple geometric problems.

Chapter lessons

3-1. Equation of a circle

This lesson defines the circle algebraically as the locus of points equidistant from a centre. We derive the standard and general forms of the equation in the Cartesian plane. Mastering these forms is essential for all subsequent analysis.

4. Ellipses

This chapter introduces the ellipse, the second major conic section, defining it by its unique two-foci property. We derive the standard and general algebraic equations for the curve. Mastering these equations is essential for modelling satellite orbits and planetary motion. By the end of this chapter, you will master: defining the ellipse using its foci and locus property; deriving the standard non-parametric equation; identifying key parameters like major/minor axes; and manipulating the equation for general problem-solving.

Chapter lessons

4-1. Equation of an ellipse

This lesson defines the ellipse algebraically as the locus of points where the sum of the distances from two fixed foci is constant . We derive both the standard and general forms of the equation. Understanding these forms is essential for applications in orbital mechanics.

5. Parabolas

This chapter establishes the parabola as a fundamental conic section by defining its equation algebraically in the $x$-$y$ plane. We derive the standard forms using the focus and directrix. Understanding this unique curve is essential for modelling trajectories in physics and designing reflector systems in engineering. By the end of this chapter, you will master: defining the parabola using its geometric properties; deriving the standard non-parametric equation; analysing the relationship between the vertex and focus; and applying the parametric form to describe the curve.

Chapter lessons

5-1. Equation of a parabola

This lesson defines the parabola as the locus of points equidistant from a fixed focus and directrix . We derive the standard non-parametric equation and introduce its parametric form. Mastering these forms is essential for applications in projectile motion and optics.

6. Hyperbolas

This chapter defines the hyperbola, the final and most complex of the conic sections, based on the difference between distances from its two foci. We derive the standard algebraic and parametric forms of its equation. Mastery of this curve is crucial for understanding specific celestial paths and navigation systems. By the end of this chapter, you will master: defining the hyperbola using its locus property; deriving the standard non-parametric equation; identifying key parameters including asymptotes and eccentricity; and understanding the parametric form of the curve.

Chapter lessons

6-1. Equation of a hyperbola

This lesson defines the hyperbola as the locus of points where the difference between the distances from two fixed foci is constant . We derive both the standard non-parametric equation and introduce its parametric form. This curve is essential for celestial mechanics and navigation systems.

7. Tangents and Normals

This chapter applies differential calculus to analyse curves. We define tangents as lines that locally touch a curve, and normals as lines perpendicular to them. Mastering this is essential for understanding the specific properties of any given point on a curve. By the end of this chapter, you will master: using differentiation to find the gradient of a curve; determining the equation of the tangent at a given point; and determining the equation of the normal at a given point.

Chapter lessons

7-1. Tangents

This lesson defines a tangent as a straight line that touches a curve at precisely one point. You will learn to use differentiation to find the gradient of the curve at that point. This gradient is essential for deriving the tangent's equation.

7-2. Normals

This lesson defines the normal as the line perpendicular to the tangent at a curve's specific point. You will learn to use the negative reciprocal of the tangent's gradient to determine the normal's equation. This is crucial for problems in mechanics and geometry.